When an electron in an excited atom jumps from the $L$ shell to the $K$ shell,$X$-rays are emitted. These $X$-rays undergo first-order diffraction at an angle of $7.75^{\circ}$ by a crystal with an interplanar spacing of $2.64 \ \mathring{A}$. Calculate the energy difference between the $K$ shell and the $L$ shell. $(\sin \ 7.75^{\circ} = 0.1349)$

• A
  $2.788 \times 10^{-13} \ J$
• B
  $2.788 \times 10^{-15} \ J$
• C
  $3.586 \times 10^{12} \ J$
• D
  $1.075 \times 10^{13} \ J$